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Colorful Helly-type Theorems for Ellipsoids Naszodi, Marton


We prove the following Helly-type result. Let $\mathcal{C}_1,\ldots,\mathcal{C}_{3d}$ be finite families of convex bodies in $\mathbb{R}^d$. Assume that for any colorful choice of $2d$ sets, $C_{i_k}\in \mathcal{C}_{i_k}$ for each $1\leq k\leq 2d$ with $1\leq i_1< \ldots < i_{2d}\leq 3d$, the intersection $\bigcap\limits_{k=1}^{2d} C_{i_k}$ contains an ellipsoid of volume at least 1. Then there is a color class $1\leq i \leq 3d$ such that $\bigcap\limits_{C\in \mathcal{C}_i} C$ contains an ellipsoid of volume at least $d^{-O(d^2)}$. Joint work with Gábor Damásdi and Viktória Földvári.

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